Based on historical data, your manager believes that 33% of the company's orders come from first-time customers. A random sample of 102 orders will be used to estimate the proportion of first-time-customers. What is the probability that the sample proportion is less than 0.27? Note: You should carefully round any z-values you calculate to 4 decimal places.

Answer =

(Enter your answer as a number accurate to 4 decimal places.)

Pete owns a small business renting out bicycles in Vancouver. He rents each bike for $5 a day. After a year, his rental bikes are worn out and they become basically worthless. So each year Pete gives his old bikes to a local charity and replaces his entire fleet of bikes. Pete buys his bikes from a local manufacturer at the discounted price of $500 each.
Pete has estimated the following probability distribution of the number of bikes he can expect to rent on any given day:

After Pete rents out a bike, he has to clean and tune it before he can rent it out again. The average cost of this maintenance is $2 per bike. This includes the cost of materials and the opportunity cost of Pete’s time.
Pete’s store is open five days a week year round, except during a four-week period each January when he goes skiing.
How many bikes should Pete buy each year to maximize his expected profit?

. Every Tuesday afternoon during the school year, a certain university brought in a visiting speaker to present a lecture on some topic of current interest. On the day after the fourth lecture of the year, a random sample 250 students was selected from the student body at the university, and each of these students was asked how many of the four lectures they had attended. The counts for each combination of number of lectures and classification are given in the table below. number of lectures attended 0 1 2 3 4 freshman 14 19 20 4 13 classification sophomore 10 16 27 6 11 junior 15 15 17 4 9 senior 19 8 6 5 12 Suppose that an student is selected at random from this group. Let A denote the event that the selected student is a freshman, B denote the event that the selected student attended 3 lectures.

a) Calculate P(A), P(B), and P(A ∩ B).

b) Calculate both P(A | B) and P(B | A), and explain, in context, what each of these probabilities represents.

c) Calculate the probability that the selected student attended at least 2 lectures.

d) If the selected student attended at least 2 lectures, what is the probability that he or she is a junior?

A certain large shipment comes with a guarantee that it contains no more than 25% defective items. If the proportion of items in the shipment is greater than 25%, the shipment may be returned. You draw a random sample of 12 items and test each one to determine whether it is defective.

a. If in fact 25% of the items in the shipment are defective (so that the shipment is good, but just barely) what is the probability that 7 or more of the 12 sampled items are defective?

b. Based on the answer to part (a), if 25% of the items in the shipment are defective would 7 defectives in a sample of size 12 be an unusually large number?

c. If you found that 7 of the 12 sample items were defective, would this be convincing evidence that the shipment should be returned?

d. If in fact 25% of the items in the shipment are defective, what is the probability that 2 or more of the 12 sampled items are defective?

e. Based on the answer to part (d), if 25% of the items in the shipment are defective, would 2 defectives in a sample of 12 be an unusually large number?

f. If you found that 2 of 12 sample items were defective, would this be convincing evidence that the shipment should be returned? Explain.

g. Find the mean,

h. Find the variance,

i. Find the standard deviation,

**SHOW ALL WORK IN EXCEL QM**

Problem-5:

In the previous problem suppose the sale of football programs described by the probability distribution only applies to days when the weather is good. When poor weather occurs on the day of a football game, the crowd that attends the game is only half of capacity. When this occurs, the sales of programs decreases, and the total sales are given in the following table:

Programs must be printed two days prior to game day. The university is trying to establish a policy for determining the number of programs to print based on the weather forecast.

1. If the forecast is for a 20% chance of bad weather, simulate the weather for ten games with this forecast.
2. Simulate the demand for programs at 10 games in which the weather is bad.
3. Beginning with a 20% chance of bad weather and an 80% chance of good weather, develop a flowchart that would be used to prepare a simulation of the demand for football programs for 10 games.
4. Suppose there is a 20% chance of bad weather, and the university has decided to print 2,500 programs. Simulate the total profits that would be achieved for 10 football games.

Problem 15-27 Gubser Welding, Inc., operates a welding service for construction and automotive repair jobs. Assume that the arrival of jobs at the company's office can be described by a Poisson probability distribution with an arrival rate of two jobs per 8-hour day. The time required to complete the jobs follows a normal probability distribution, with a mean time of 3.2 hours and a standard deviation of 2 hours. Answer the following questions, assuming that Gubser uses one welder to complete all jobs:

What is the mean arrival rate in jobs per hour? If required, round your answer to two decimal places.

What is the mean service rate in jobs per hour? If required, round your answer to four decimal places.

What is the average number of jobs waiting for service? If required, round your answer to three decimal places.

What is the average time a job waits before the welder can begin working on it? If required, round your answer to one decimal place.

What is the average number of hours between when a job is received and when it is completed? If required, round your answer to one decimal place. hours

What percentage of the time is Gubser's welder busy?

1. This problem has two parts. BUT the second part will appear ONLY AFTER you have answered the first part correctly.)

Rework problem 9 in section 4.2 of your text, involving a defective vending machine. Assume that the machine yields the item selected 70 percent of the time, and returns nothing 30 percent of the time. Three individuals attempt to use the machine. Let XX be defined as the number of individuals who obtain the item selected.

2.

Rework problem 16 in section 4.2 of your text, involving drawing markers from a box of markers with ink and markers without ink. Assume that the box contains 18 markers: 13 that contain ink and 5 that do not contain ink. A sample of 7 markers is selected and a random variable YY is defined as the number of markers selected which do not have ink.

How many different values are possible for the random variable YY?

Fill in the table below to complete the probability density function. Be certain to list the values of YYin ascending order.

How many different values are possible for the random variable XX?

3.

Fill in the table below to complete the probability density function. Be certain to list the values of XXin ascending order.

Innocent until proven guilty? In Japanese criminal trials, about 95% of the defendants are found guilty. In the United States, about 60% of the defendants are found guilty in criminal trials†. Suppose you are a news reporter following six criminal trials.

(a) If the trials were in Japan, what is the probability that all the defendants would be found guilty? (Round your answer to three decimal places.)


What is this probability if the trials were in the United States? (Round your answer to three decimal places.)



(b) Of the six trials, what is the expected number of guilty verdicts in Japan? (Round your answer to two decimal places.)
verdicts

What is the expected number in the United Sates? (Round your answer to two decimal places.)
verdicts

What is the standard deviation in Japan? (Round your answer to two decimal places.)
verdicts

What is the standard deviation in the United States? (Round your answer to two decimal places.)
verdicts


(c) As a U.S. news reporter, how many trials n would you need to cover to be at least 99% sure of two or more convictions?
trials

How many trials n would you need if you covered trials in Japan?
trials

According to the Kentucky Transportation Cabinet, an average of 167,000 vehicles per day crossed the Brent Spence Bridge into Ohio in 2009. Give the state of disrepair the bridge is currently under, a journalist would like to know if the mean traffic count has increased over the past five years. Assume the population of all traffic counts is bimodal with a standard deviation of 15,691 vehicles per day.
a. What conjecture would the journalist like to find support for in this sample of vehicles?
b. A sample of 75 days is taken and the traffic counts are recorded, completely describe the sampling distribution of the sample mean number of vehicles crossing the Brent Spence Bridge. Type out all supporting work.
c. The sample of 75 days had an average of 172,095.937 vehicles crossing the bridge. What is the probability of observing a sample mean of 172,095.937 vehicles or larger? Type out all supporting work.
d. Based on the probability computed in part c, what can be conclusion can be made about the conjecture? Explain.
e. If the number of sampled days was changed to 25, how would the shape, mean, and standard deviation of the sampling distribution of the sample mean traffic counts be affected?

A class has 40 students.

• Thirty students are prepared for the exam,

• Ten students are unprepared. The professor writes an exam with 10 questions, some are hard and some are easy.

• 7 questions are easy. Based on past experience, the professor knows that: – Prepared students have a 90% chance of answering easy questions correctly – Unprepared students have a 50% chance of answering easy questions correctly.

• 3 questions are hard. Based on past experience, the professor knows that: – Prepared students have a 50% chance of answering hard questions correctly – Unprepared students have a 10% chance of answering hard questions correctly

• Each student’s performance on each question is independent of their performance on other questions.

(a) Find the probability that a prepared student answers all 10 questions correctly.

(b) What is the probability that at least one of the 30 prepared students answers all 10 questions correctly. Assume that each student’s score is independent of every other student.

(c) Let P be the number of questions answered correctly by a randomly chosen prepared student, and let U be the number answered correctly by a randomly chosen unprepared student. Find E[P] and E[U]

(d) Find Var(P) and Var(U)